There is no odd perfect polynomial over $\mathbb F_2$ with four prime factors
Luis H. Gallardo, Olivier Rahavandrainy · Portugaliae Mathematica · 2009
A perfect polynomial over the binary field \mathbb F_2 is a polynomial A ∈ \mathbb F_2[x] that equals the sum of all its divisors. If \operatorname{gcd}(A,x^2 + x) = 1 then we say that A is odd. It is believed that odd perfect polynomials do not exist. In this article we prove this for odd perfect polynomials A with four prime divisors, i.e., polynomials of the form A = P^aQ^bR^cS^d where P, Q, R, S are distinct irreducible polynomials of degree > 1 over \mathbb F_2 and a, b, c, d are positive integers.